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Simple Harmonic Motion and Elasticity

Simple Harmonic Motion and Elasticity. Force is proportional to displacement. For small displacements, the force required to stretch or compress a spring is directly proportional to the displacement x, or

isaiah-myers
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Simple Harmonic Motion and Elasticity

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  1. Simple Harmonic Motion and Elasticity

  2. Force is proportional to displacement For small displacements, the force required to stretch or compress a spring is directly proportional to the displacement x, or The constant k is called the spring constant or stiffness of the spring. A spring behaves according to the above equation is said to be an ideal spring.

  3. 10.1The Ideal Spring and Simple Harmonic Motion spring constant Units: N/m

  4. 10.1 The Ideal Spring and Simple Harmonic Motion Example-1: A Tire Pressure Gauge The spring constant of the spring is 320 N/m and the bar indicator extends 2.0 cm. What force does the air in the tire apply to the spring?

  5. 10.1 The Ideal Spring and Simple Harmonic Motion

  6. Tire Pressure? Tire pressure varies from 2 bar to 2.2 bar. Or, 28 psi to 32 psi 1 bar = 100 kPa = 100 000 Pa = 100 000 N/m2 Pressure = Force per unit Area P = F/A What is the area of the Tire Pressure Gauge?

  7. 10.1 The Ideal Spring and Simple Harmonic Motion Conceptual Example-2: Are Shorter Springs Stiffer? A 10-coil spring has a spring constant k. If the spring is cut in half, so there are two 5-coil springs, what is the spring constant of each of the smaller springs?

  8. Shorter Springs are Stiffer The spring constant of each 5-coil spring is 2k. Spring constant  1/# of coils k for 10 coils 2k for 5 coils 10k for 1 coil

  9. Restoring Force To stretch or compress a spring, a force must be applied to it. In accord with Newton’s third law, the spring exert an oppositely directed force of equal magnitude. This reaction force is applied by the spring to the agent that does the pulling or pushing. The reaction force is also called a “restoring force”. Fx = - kx The reaction force always points ina direction opposite to the displacement of the spring from its unstrained length.

  10. Restoring Force The restoring force of a spring can also contribute to the net external force.

  11. 10.1 The Ideal Spring and Simple Harmonic Motion HOOKE’S LAW: RESTORING FORCE OF AN IDEAL SPRING The restoring force on an ideal spring is

  12. Why it is called Restoring Force An object of mass m is attached to a spring on a frictionless table. In part A, the spring has been stretched to the right, so the spring exerts the left-ward pointing force Fx. When the object is released, this force pulls it to the left, restoring it toward its equilibrium position.

  13. Restoring Force causes Simple Harmonic Motion When the object is released, the restoring force pulls it to its equilibrium position. In accord with Newton’s first law, the moving object has inertia and coasts beyond the equilibrium position, compressing the spring as in part B. The restoring force exerted by the spring now points to the right and after bringing the object to a momentary halt, acts to restore the object to its equilibrium position. Since no friction acts on the object the back-and-forth motion repeats itself. When the restoring force has the mathematical form given by Fx = -kx, the type of friction-free motion is designated as “simple harmonic motion”.

  14. Simple Harmonic Motion By attaching a pen to the object and moving a strip of paper past it at a steady rate, we can record the position of the vibrating object as time passes. The shape of this graph is characteristic of simple harmonic motion and is called sinusoidal.

  15. Simple Harmonic Motion Simple harmonic motion, like any motion, can be described in terms of • displacement, • velocity, and • acceleration.

  16. 10.2 Simple Harmonic Motion and the Reference Circle DISPLACEMENT

  17. 10.2 Simple Harmonic Motion and the Reference Circle Radius = A The displacement of the shadow, x is just the projection of the radius A onto the x-axis: Where  = t , the angular speed in rad/s

  18. Angular Speed,   = /t rad/s • = t rad Angular displacement () for one cycle is 2 rad in T. So,  = 2/T  = 2 Because  is directly proportional to the frequency ,  is often called the angular frequency.

  19. 10.2 Simple Harmonic Motion and the Reference Circle amplitude A: the maximum displacement period T: the time required to complete one cycle frequency f: the number of cycles per second (measured in Hz)

  20. 10.2 Simple Harmonic Motion and the Reference Circle VELOCITY Tangential velocity, VT = Radius x angular velocity Velocity of the shadow, Vx = X component of VT Maximum Velocity of the shadow, Vx = A

  21. 10.2 Simple Harmonic Motion and the Reference Circle • Example 3 The Maximum Speed of a Loudspeaker Diaphragm • The frequency of motion is 1.0 KHz and the amplitude is 0.20 mm. • What is the maximum speed of the diaphragm? • Where in the motion does this maximum speed occur?

  22. 10.2 Simple Harmonic Motion and the Reference Circle (a) • The maximum speed • occurs midway between • the ends of its motion.

  23. Simple harmonic motion? • Is the motion of the lighted bulb simple harmonic motion, when each lights for 0.5s in sequence?

  24. 10.2 Simple Harmonic Motion and the Reference Circle ACCELERATION The ball on the reference circle moves in uniform circular motion, and, therefore, has centripetal acceleration ac that points toward the centre of the circle. The acceleration ax of the shadow is the x component of the centripetal acceleration ac .

  25. Maximum Acceleration • A loudspeaker vibrating at 1kHz with an amplitude of 0.20mm has a maximum acceleration of amax = A2 = 7.9 x 103 m/s2 • Maximum acceleration occurs when the force acting on the diaphragm is a maximum. • The maximum force arises when the diaphragm is at the ends of its path.

  26. Frequency of Vibration • With the aid of the Newton’s second law , it is possible to determine the frequency at which an object of mass m vibrates on a spring. • Mass of the spring is negligible • Only force acting is the restoring force.

  27. 10.2 Simple Harmonic Motion and the Reference Circle FREQUENCY OF VIBRATION Larger spring constants k and smaller masses m result in larger frequencies.

  28. 10.2 Simple Harmonic Motion and the Reference Circle Example 6 A Body Mass Measurement Device The device consists of a spring-mounted chair in which the astronaut sits. The spring has a spring constant of 606 N/m and the mass of the chair is 12.0 kg. The measured period is 2.41 s. Find the mass of the astronaut.

  29. 10.2 Simple Harmonic Motion and the Reference Circle

  30. 10.3 Energy and Simple Harmonic Motion A compressed spring can do work.

  31. Average magnitude of Force Spring force at x0 is kx0 Spring force at xf is kxf Average Fx = ½(kx0+kxf)

  32. 10.3 Energy and Simple Harmonic Motion Work done by the average spring force: Final elastic PE Initial elastic PE

  33. 10.3 Energy and Simple Harmonic Motion DEFINITION OF ELASTIC POTENTIAL ENERGY The elastic potential energy is the energy that a spring has by virtue of being stretched or compressed. For an ideal spring, the elastic potential energy is SI Unit of Elastic Potential Energy:joule (J)

  34. 10.3 Energy and Simple Harmonic Motion Conceptual Example 8 Changing the Mass of a Simple Harmonic Oscilator The box rests on a horizontal, frictionless surface. The spring is stretched to x=A and released. When the box is passing through x=0, a second box of the same mass is attached to it. Discuss what happens to the (a) maximum speed (b) amplitude (c) angular frequency.

  35. Doubling the Mass of a Simple Harmonic Oscillator • At x=0m, a second box of the same mass and speed vmax is attached. • So, the max KE is doubled as mass is doubled. • So, when the spring compresses it will have double the PEe . • As PEe is doubled max amplitude will be 2 times

  36. Doubling the Mass causes max amplitude to 2 times • Before adding the second mass, the displacement x1 is related to the PE as: • As PEe is doubled the new amplitude is x2

  37. Doubling the Mass of a Simple Harmonic Oscillator Angular frequency is So, a doubling of mass will cause the angular frequency reduce to (1/ 2 ) 

  38. 10.3 Energy and Simple Harmonic Motion Example 9 A falling ball on a vertical Spring A 0.20-kg ball is attached to a vertical spring. The spring constant is 28 N/m. When released from rest, how far does the ball fall before being brought to a momentary stop by the spring?

  39. 10.3 Energy and Simple Harmonic Motion

  40. 10.4 The Pendulum A simple pendulum consists of a particle mass m attached to a frictionless pivot by a cable of negligible mass.

  41. simple pendulum The force of gravity is responsible for the back-and-forth rotation about the axis at P. A net torque is required to change the angular speed. The gravitational force mg produces the torque. • = - (mg)l - sign to represent the restoring force • = - (mg)L • = - k’  where k’ = mgL • Same as F = - kx So,  = (k’/m) = (mgL/m) = (mgL/I) in rotational motion in place of mass moment of inertia ‘I’ will appear

  42. simple pendulum So,  = (k’/m) = (mgL/m) = (mgL/I) in rotational motion in place of mass moment of inertia ‘I’ will appear The moment of inertia of a particle of mass m, rotating at a radius L about an axis is I = mL2 Frequency of a simple pendulum is Which does not depend on the mass of the particle.

  43. 10.4 The Pendulum Example 10 Keeping Time Determine the length of a simple pendulum that will swing back and forth in simple harmonic motion with a period of 1.00 s.

  44. 10.5 Damped Harmonic Motion In simple harmonic motion, an object oscillated with a constant amplitude. In reality, friction or some other energy dissipating mechanism is always present and the amplitude decreases as time passes. This is referred to as damped harmonic motion.

  45. 10.5 Damped Harmonic Motion • simple harmonic motion • 2 & 3) underdamped • critically damped • 5) overdamped

  46. 10.6 Driven Harmonic Motion and Resonance When a force is applied to an oscillating system at all times, the result is driven harmonic motion. Here, the driving force has the same frequency as the spring system and always points in the direction of the object’s velocity.

  47. 10.6 Driven Harmonic Motion and Resonance RESONANCE Resonance is the condition in which a time-dependent force can transmit large amounts of energy to an oscillating object, leading to a large amplitude motion. Resonance occurs when the frequency of the force matches a natural frequency at which the object will oscillate.

  48. Elastic Deformation • All materials become distorted when they are squeezed or stretched. • Those materials return to their original shape when the deforming force is removed, such materials are called “elastic”.

  49. 10.7 Elastic Deformation Atomic View of Elastic materials Because of these atomic-level “springs”, a material tends to return to its initial shape once forces have been removed. ATOMS FORCES

  50. Stretching force • Magnitude of the deforming force can be expressed as follows, provided the amount of stretch or compression is small compared to the original length of the object.

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